Like most mathematics students I have also spent a considerable amount of time on trying to solve the Collatz-conjecture. It is a fascinating problem and one I would love to see the solution of within my lifetime. It defines a simple function mapping a positive integer to another positive integer. It goes like this: \[ f(n) = \begin{cases} \frac{n}{2}\ &n \equiv 0 \quad \text{mod}(2) \\ 3n+1 &n \equiv 1 \quad \text{mod}(2) \end{cases} \] Applying this function repeatedly, starting with the number \(1\) we receive \(1 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 4 \rightarrow \; ...\). The conjecture is, that for any positive integer applying \(f\) repeatedly will eventually produce the number \(1\) and thus end up in this infinite cycle. Here's a few things I learned during my time with this problem:

1. There is a close relationship between this problem and the so-called ruler function. In the graph it is drawn in red.
2. Expanding on this, it's worthwhile to not just consider the sequence of numbers, but also the divisibility by 2 when you add one to the number (drawn in blue). Using this it's for example possible to proof the conjecture for the modified function with \(f(n) = 3n\) if \(n\) is odd.
3. Trying to solve an impossible problem is a great way to stumble upon all sorts of methods for solving other problems.

References

• Collatz Conjecture Wikipedia Article